Subject: [mg52154] Re: Fibonacci based sum that is b-normal on binary numbers

From: Peter Pein <petsie at arcor.de>

Date: Fri, 12 Nov 2004 02:14:22 -0500 (EST)

References: <cmve7h$sh9$1@smc.vnet.net>

Sender: owner-wri-mathgroup at wolfram.com

Roger Bagula wrote:
> This sum and it's b-normal sequence is due
> to work of a friend who doesn't like me to use his name here or elsewhere
> ..He came up with two very nice sums using Fibonacci numbers.
> I used the Binet function in them and got very good agreement.
> So I tried them in a b-normal.
> I had to modify the result some to get this result.
> I get a new sum that appears irrational
> and an iteration that is b-normal .
> I think that using the Binet function in this makes it
> a new sequence sum.
> I thought that this was a very remarkable result.
>
> Clear[x,a,digits,f,fib]
> (* convergent sum based on Fibonacci sequence to make a binary b-normal
> iteration *)
> digits=200
> fib[n_Integer?Positive] :=fib[n] = fib[n-1]+fib[n-2]
> fib[0]=0;fib[1] = fib[2] = 1;
> sfib=Sum[fib[n]/((n+1)*2^(n+1)),{n,0,digits}]
> N[sfib,digits]
> x[n_]:=x[n]=Mod[2*x[n-1]+fib[n-1]/(2*n),1]
> x[0]=0
> a=Table[N[x[n],digits],{n,0,digits}]
> ListPlot[a,PlotJoined->True,PlotRange->All]
> b=Sort[Table[N[x[n],digits],{n,0,digits}]];
> ListPlot[b,PlotJoined->True,PlotRange->All]
> Respectfully, Roger L. Bagula
>
> tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
> alternative email: rlbtftn at netscape.net
> URL : http://home.earthlink.net/~tftn
>
...just another way to compute
sfib = 1/10*(5*Log[4] + Sqrt[5]*Log[1/2*(7 - 3*Sqrt[5])])
--
Peter Pein
Berlin